Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&, \\ P_{\mathbf{5}}&{}\approx{}&0.988436245040&{}:{}&0.000266534449&{}:{}&0.011297220510&, \\ P^-_{\mathbf{5}}&{}\approx{}&1.087225286554&{}:{}&-0.047787429778&{}:{}&-0.039437856776&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.908795282426&{}:{}&0.039006296572&{}:{}&0.052198421002&, \\ Q_{\mathbf{5}}&{}\approx{}&0.827079539377&{}:{}&-0.024796420549&{}:{}&0.197716881172&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.834847233084&{}:{}&0.021238066511&{}:{}&0.143914700405&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{5}}\)
\(P^-_{\mathbf{5}}\)
\(P^+_{\mathbf{5}}\)
\(Q_{\mathbf{5}}\)
\(I^\prime_{\mathbf{5}}\)
5 (202)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.915631799079&{}:{}&0.040496736442&{}:{}&0.043871464479&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.823111595205&{}:{}&-1.963201742740&{}:{}&2.140090147534&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.698502526124&{}:{}&1.676406062698&{}:{}&-1.374908588822&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.755704774914&{}:{}&0.005630791516&{}:{}&0.238664433570&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.910510351933&{}:{}&0.079083072757&{}:{}&0.010406575310&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.913773251376&{}:{}&0.000246401376&{}:{}&0.085980347248&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.298199128586&{}:{}&0.711234117383&{}:{}&0.586965011203&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.992108441892&{}:{}&0.043879159869&{}:{}&-0.035987601761&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.996055317793&{}:{}&-0.043780184423&{}:{}&0.047724866630&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.143397840021&{}:{}&1.143397840021&,\\B^*_{\mathbf{5}}&{}\approx{}&0.807067162602&{}:{}&0.000000000000&{}:{}&0.192932837398&,\\C^*_{\mathbf{5}}&{}\approx{}&1.030907319333&{}:{}&-0.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.101241841105&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&4.938669571233&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&4.191015156744&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.915631799079&{}:{}&0.040496736442&{}:{}&0.043871464479&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.823111595205&{}:{}&-1.963201742740&{}:{}&2.140090147534&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.698502526124&{}:{}&1.676406062698&{}:{}&-1.374908588822&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.988436245040&{}:{}&0.000266534449&{}:{}&0.011297220510&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.755704774914&{}:{}&0.005630791516&{}:{}&0.238664433570&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.910510351933&{}:{}&0.079083072757&{}:{}&0.010406575310&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.913773251376&{}:{}&0.000246401376&{}:{}&0.085980347248&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&1.087225286554&{}:{}&-0.047787429778&{}:{}&-0.039437856776&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.298199128586&{}:{}&0.711234117383&{}:{}&0.586965011203&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.992108441892&{}:{}&0.043879159869&{}:{}&-0.035987601761&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.996055317793&{}:{}&-0.043780184423&{}:{}&0.047724866630&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.908795282426&{}:{}&0.039006296572&{}:{}&0.052198421002&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.915631799079&{}:{}&0.040496736442&{}:{}&0.043871464479&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.823111595205&{}:{}&-1.963201742740&{}:{}&2.140090147534&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.698502526124&{}:{}&1.676406062698&{}:{}&-1.374908588822&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.755704774914&{}:{}&0.005630791516&{}:{}&0.238664433570&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.910510351933&{}:{}&0.079083072757&{}:{}&0.010406575310&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.913773251376&{}:{}&0.000246401376&{}:{}&0.085980347248&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.827079539377&{}:{}&-0.024796420549&{}:{}&0.197716881172&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.143397840021&{}:{}&1.143397840021&,\\B^*_{\mathbf{5}}&{}\approx{}&0.807067162602&{}:{}&0.000000000000&{}:{}&0.192932837398&,\\C^*_{\mathbf{5}}&{}\approx{}&1.030907319333&{}:{}&-0.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.834847233084&{}:{}&0.021238066511&{}:{}&0.143914700405&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.755704774914&{}:{}&0.005630791516&{}:{}&0.238664433570&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.910510351933&{}:{}&0.079083072757&{}:{}&0.010406575310&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.913773251376&{}:{}&0.000246401376&{}:{}&0.085980347248&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.143397840021&{}:{}&1.143397840021&,\\B^*_{\mathbf{5}}&{}\approx{}&0.807067162602&{}:{}&0.000000000000&{}:{}&0.192932837398&,\\C^*_{\mathbf{5}}&{}\approx{}&1.030907319333&{}:{}&-0.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
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